Algebra of Complex Numbers

Q.

Reduce $[\frac{1}{1-4i}-\frac{2}{1+i}]=\left[\frac{3-4i}{5+i}\right]$ to the standard form.

Q.

If ${}^{22}{P}_{r+1}{:}^{20}{P}_{r+2}=11:52,$ find r.

Q.

Express the complex numbers in the form of a + ib:

$i^{-39}$

Q.

Express the complex numbers in the form of a + ib:

$(1-i{)}^4$

Q.

Express the complex numbers in the form of a + ib:

$3(7+i7)+i(7+i7)$

Q. Let 2x+3y+4z = 9, x, y, z > 0 then find the maximum value of (1+x)}^{2 }(2+y)}^3(4+z)}^4

Q. If (1+i)(1+2i)(1+3i)......(1+ni) = a+ib, then 2.5.10.....(1+$n^2$) is equal to

1. 

2. 

3. 

4. 

Q. The equation $\overline{b}z$+$\overline{z}b=c$ where b is a non-zero complex constant and c is real, represents a

1. circle
2. straight line
3. ellipse
4. hyperbola

Q. Express $(-\sqrt{3}+\sqrt{-2})(2\sqrt{3}-i)$ in the form (a+ib).

Q.

The sum of an infinite G.P. is 23 and the sum of squares of the series is 69, then first term is :

1. 69/2

2. 10/3

3. 9/10

4. 13/69

Q.

Prove that:
(i) $i^n+i^{n+1}+i^{n+2}+i^{n+3}=0$
(ii) $i^{107}+i^{112}+i^{117}+i^{122}=0$
(iii) $(1+i{)}^4\times {(1+\frac{1}{i})}^4=16$

Q.

Express the complex numbers in the form of a + ib:

$\left(5i\right)(-\frac{3}{5}i)$

Q. If $z_1,z_2\in C,z_1^2+z_2^2\in R,z_1(z_1^2-3z_2^2)=2$ and $z_2(3z_1^2-z_2^2)=11$, then the value of $z_1^2+z_2^2$ is

1. $10$
2. $12$
3. $5$
4. $8$

Q. the sum of 1+2/5 +3/5^2 +4/5^{3 } upto n terms i

Q.

If the sum of three numbers in A.P. is 12 and sum of their cubes is 408, then sum of their squares is:

1. 144

2. 66

3. 36

4. None

Q.

If x=-5+2$\sqrt{-4}$, then the values of the expression

$x^4$+9$x^3$+35$x^2$-x+4 is

1. -160

2. 60

3. 160

4. -60

Q. If $z=(-\sqrt{3}+\sqrt{-2})(2\sqrt{3}-i)$, then

1. $Im\left(z\right)=\sqrt{3}(1-2\sqrt{2})$
2. $Re\left(z\right)=(-6-\sqrt{2})$
3. $Im\left(z\right)=\sqrt{3}(1+2\sqrt{2})$
4. $Re\left(z\right)=(-6+\sqrt{2})$

Q. The maximum and minimum value for the function y=x^3-3x^2+6 are

Q.

Find the sum of the infinite G.P. 2, $\frac{2}{3}$, $\frac{2}{9}$, $\frac{2}{27}$ . . . . .

__

Q. If $z$ is a non-zero complex number, then the area of the quadrilateral formed by the points $z,\overline{z},-z$ and $-\overline{z}$ is

1. $|z{|}^2$
2. $4|z{|}^2$
3. $|z^2+{\overline{z}}^2|$
4. $|z^2-{\overline{z}}^2|$

Q.

If $\frac{c+i}{c-i}$ = a+ib, where a, b, c are real, then $a^2+b^2$ =

1. 1

2. -1

3. $c^2$

4. $-c^2$

Q.

(i) If ${}^n{P}_4{:}^n{P}_5=1:2$ find n.

(ii) If ${}^{n-1}{P}_3{:}^{n+1}=5:12,$ find n.

Q. lim x tend to 2 2^x+8-1024/4^x-16

Q.

Express the complex numbers in the form of a + ib:

${(-2-\frac{1}{3}i)}^3$

Q.

If $\frac{c+i}{c-i}$ = a+ib, where a, b, c are real, then $a^2+b^2$ =

1. -1

2. 

3. 1

4. 

Q. If , then find the least positive integral value of m .

Q.

If (a+ib)(c+id)(e+if)(g+ih)=A+iB, then ($a^2$+$b^2$)($c^2$+$d^2$)($e^2$+$f^2$)($g^2$+$h^2$)=

1. A²+B²

2. A²-B²

3. A²

4. B²

Q. Reduce $(\frac{1}{1-4i}-\frac{2}{1+i})\left(\frac{3-4i}{5+i}\right)$ to the standard form.

Q. For any two complex numbers z 1 and z 2 , prove that Re (z 1 z 2 ) = Re z 1 Re z 2 – Im z 1 Im z 2

Q. calculas differentiation 1) d/dx = ln(10x^3+6) 2) y=e^(2t^3+6) find dy/dt